Connected co-spectral graphs are not necessarily both Hamiltonian
Filar, Jerzy A
Lucas, Stephen K
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There are many examples of co-spectral graphs where one is connected while the other is not. A Hamiltonian cycle is a closed path that visits every vertex exactly once. Naturally, a graph needs to be connected to be Hamiltonian. We are not aware of any statements in the literature relating co-spectral graphs and Hamiltonicity.